And there is the matter of a lot more braid between Kirk and Rand.
Not only that, but she was his yeoman, his personal assistant and secretary, working directly under him. She was the Radar to his Colonel Potter. So it's a particularly close working relationship that has to go smoothly if he wants to do his job effectively. Even if he were free to romance other members of his crew, it would be a bad idea with his own yeoman.
Tomlinson was a lieutenant, and was stated in dialogue to be Martine's superior officer.
Scotty referred to him in dialogue as "Specialist Tomlinson." James Blish's adaptation refers to the couple at the wedding as "Specialist (phaser) Robert Tomlinson and Spec. 2nd Cl. (phaser) Angela Martine," most likely taken verbatim from the script.
Specialist is an enlisted rating; at the time TOS was made, it was used in the U.S. Navy as part of the petty officer pay structure.
Yes, Tomlinson has a lieutenant's stripe on his uniform, which is the reason Memory Alpha jumps to the conclusion that he's a lieutenant, but there are many instances in Trek of erroneous costuming with regard to rank, notably Chief O'Brien being given lieutenant's pins for quite some time before an enlisted pin was designed.
So Memory Alpha is wrong. Tomlinson and Martine were both enlisted petty officers one step apart in rating. As I said, I don't think ranks/ratings matter as much for enlisted personnel as for officers.
That interpretation of Kirk being just a walking gland really bugs me.
Indeed. Kirk was originally written as just the opposite -- in "Mudd's Women," he was the only human male on the ship unaffected by the title characters. Most of his romantic plots were either old flames from the past, the result of an altered mental state, women chasing after him with mixed results, calculated seductions in service to the mission, or in rare cases, sincere love affairs (as with Edith or Miramanee). In subsequent seasons, he was written closer to the womanizing norm for TV action heroes of the day, but Kirk was practically a monk compared to womanizers like
The Wild Wild West's Jim West or
The Man from UNCLE's Napoleon Solo.